src/Pure/library.ML
author haftmann
Tue Jun 05 15:17:02 2007 +0200 (2007-06-05)
changeset 23251 471b576aad25
parent 23220 9e04da929160
child 23424 d0580634f128
permissions -rw-r--r--
moved generic algebra modules
     1 (*  Title:      Pure/library.ML
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Author:     Markus Wenzel, TU Muenchen
     5 
     6 Basic library: functions, options, pairs, booleans, lists, integers,
     7 strings, lists as sets, balanced trees, orders, current directory, misc.
     8 
     9 See also General/basics.ML for the most fundamental concepts.
    10 *)
    11 
    12 infix 1 |>>>
    13 infix 2 ?
    14 infix 3 o oo ooo oooo
    15 infix 4 ~~ upto downto
    16 infix orf andf \ \\ mem mem_int mem_string union union_int
    17   union_string inter inter_int inter_string subset subset_int subset_string
    18 
    19 signature BASIC_LIBRARY =
    20 sig
    21   (*functions*)
    22   val I: 'a -> 'a
    23   val K: 'a -> 'b -> 'a
    24   val flip: ('a -> 'b -> 'c) -> 'b -> 'a -> 'c
    25   val curry: ('a * 'b -> 'c) -> 'a -> 'b -> 'c
    26   val uncurry: ('a -> 'b -> 'c) -> 'a * 'b -> 'c
    27   val |>>> : ('a * 'c) * ('a -> 'b * 'd) -> 'b * ('c * 'd)
    28   val ? : bool * ('a -> 'a) -> 'a -> 'a
    29   val oo: ('a -> 'b) * ('c -> 'd -> 'a) -> 'c -> 'd -> 'b
    30   val ooo: ('a -> 'b) * ('c -> 'd -> 'e -> 'a) -> 'c -> 'd -> 'e -> 'b
    31   val oooo: ('a -> 'b) * ('c -> 'd -> 'e -> 'f -> 'a) -> 'c -> 'd -> 'e -> 'f -> 'b
    32   val funpow: int -> ('a -> 'a) -> 'a -> 'a
    33 
    34   (*exceptions*)
    35   exception EXCEPTION of exn * string
    36   val do_transform_failure: bool ref
    37   val transform_failure: (exn -> exn) -> ('a -> 'b) -> 'a -> 'b
    38   datatype 'a result = Result of 'a | Exn of exn
    39   val capture: ('a -> 'b) -> 'a -> 'b result
    40   val release: 'a result -> 'a
    41   val get_result: 'a result -> 'a option
    42   val get_exn: 'a result -> exn option
    43 
    44   (*errors*)
    45   exception SYS_ERROR of string
    46   val sys_error: string -> 'a
    47   exception ERROR of string
    48   val error: string -> 'a
    49   val cat_error: string -> string -> 'a
    50   val assert_all: ('a -> bool) -> 'a list -> ('a -> string) -> unit
    51 
    52   (*pairs*)
    53   val pair: 'a -> 'b -> 'a * 'b
    54   val rpair: 'a -> 'b -> 'b * 'a
    55   val fst: 'a * 'b -> 'a
    56   val snd: 'a * 'b -> 'b
    57   val eq_fst: ('a * 'c -> bool) -> ('a * 'b) * ('c * 'd) -> bool
    58   val eq_snd: ('b * 'd -> bool) -> ('a * 'b) * ('c * 'd) -> bool
    59   val eq_pair: ('a * 'c -> bool) -> ('b * 'd -> bool) -> ('a * 'b) * ('c * 'd) -> bool
    60   val swap: 'a * 'b -> 'b * 'a
    61   val apfst: ('a -> 'b) -> 'a * 'c -> 'b * 'c
    62   val apsnd: ('a -> 'b) -> 'c * 'a -> 'c * 'b
    63   val pairself: ('a -> 'b) -> 'a * 'a -> 'b * 'b
    64 
    65   (*booleans*)
    66   val equal: ''a -> ''a -> bool
    67   val not_equal: ''a -> ''a -> bool
    68   val orf: ('a -> bool) * ('a -> bool) -> 'a -> bool
    69   val andf: ('a -> bool) * ('a -> bool) -> 'a -> bool
    70   val exists: ('a -> bool) -> 'a list -> bool
    71   val forall: ('a -> bool) -> 'a list -> bool
    72   val set: bool ref -> bool
    73   val reset: bool ref -> bool
    74   val toggle: bool ref -> bool
    75   val change: 'a ref -> ('a -> 'a) -> unit
    76   val setmp: 'a ref -> 'a -> ('b -> 'c) -> 'b -> 'c
    77 
    78   (*lists*)
    79   exception UnequalLengths
    80   val single: 'a -> 'a list
    81   val the_single: 'a list -> 'a
    82   val singleton: ('a list -> 'b list) -> 'a -> 'b
    83   val apply: ('a -> 'a) list -> 'a -> 'a
    84   val foldr1: ('a * 'a -> 'a) -> 'a list -> 'a
    85   val foldl_map: ('a * 'b -> 'a * 'c) -> 'a * 'b list -> 'a * 'c list
    86   val flat: 'a list list -> 'a list
    87   val unflat: 'a list list -> 'b list -> 'b list list
    88   val burrow: ('a list -> 'b list) -> 'a list list -> 'b list list
    89   val fold_burrow: ('a list -> 'c -> 'b list * 'd) -> 'a list list -> 'c -> 'b list list * 'd
    90   val maps: ('a -> 'b list) -> 'a list -> 'b list
    91   val chop: int -> 'a list -> 'a list * 'a list
    92   val dropwhile: ('a -> bool) -> 'a list -> 'a list
    93   val nth: 'a list -> int -> 'a
    94   val nth_map: int -> ('a -> 'a) -> 'a list -> 'a list
    95   val nth_list: 'a list list -> int -> 'a list
    96   val map_index: (int * 'a -> 'b) -> 'a list -> 'b list
    97   val fold_index: (int * 'a -> 'b -> 'b) -> 'a list -> 'b -> 'b
    98   val split_last: 'a list -> 'a list * 'a
    99   val find_index: ('a -> bool) -> 'a list -> int
   100   val find_index_eq: ''a -> ''a list -> int
   101   val find_first: ('a -> bool) -> 'a list -> 'a option
   102   val get_index: ('a -> 'b option) -> 'a list -> (int * 'b) option
   103   val get_first: ('a -> 'b option) -> 'a list -> 'b option
   104   val eq_list: ('a * 'b -> bool) -> 'a list * 'b list -> bool
   105   val map2: ('a -> 'b -> 'c) -> 'a list -> 'b list -> 'c list
   106   val fold2: ('a -> 'b -> 'c -> 'c) -> 'a list -> 'b list -> 'c -> 'c
   107   val zip_options: 'a list -> 'b option list -> ('a * 'b) list
   108   val ~~ : 'a list * 'b list -> ('a * 'b) list
   109   val split_list: ('a * 'b) list -> 'a list * 'b list
   110   val separate: 'a -> 'a list -> 'a list
   111   val replicate: int -> 'a -> 'a list
   112   val multiply: 'a list -> 'a list list -> 'a list list
   113   val product: 'a list -> 'b list -> ('a * 'b) list
   114   val filter: ('a -> bool) -> 'a list -> 'a list
   115   val filter_out: ('a -> bool) -> 'a list -> 'a list
   116   val map_filter: ('a -> 'b option) -> 'a list -> 'b list
   117   val is_prefix: ('a * 'a -> bool) -> 'a list -> 'a list -> bool
   118   val take_prefix: ('a -> bool) -> 'a list -> 'a list * 'a list
   119   val chop_prefix: ('a * 'b -> bool) -> 'a list * 'b list -> 'a list * ('a list * 'b list)
   120   val take_suffix: ('a -> bool) -> 'a list -> 'a list * 'a list
   121   val prefixes1: 'a list -> 'a list list
   122   val prefixes: 'a list -> 'a list list
   123   val suffixes1: 'a list -> 'a list list
   124   val suffixes: 'a list -> 'a list list
   125 
   126   (*integers*)
   127   val inc: int ref -> int
   128   val dec: int ref -> int
   129   val upto: int * int -> int list
   130   val downto: int * int -> int list
   131   val radixpand: int * int -> int list
   132   val radixstring: int * string * int -> string
   133   val string_of_int: int -> string
   134   val signed_string_of_int: int -> string
   135   val string_of_indexname: string * int -> string
   136   val read_intinf: int -> string list -> IntInf.int * string list
   137   val read_int: string list -> int * string list
   138   val oct_char: string -> string
   139 
   140   (*strings*)
   141   val nth_string: string -> int -> string
   142   val fold_string: (string -> 'a -> 'a) -> string -> 'a -> 'a
   143   val exists_string: (string -> bool) -> string -> bool
   144   val forall_string: (string -> bool) -> string -> bool
   145   val enclose: string -> string -> string -> string
   146   val unenclose: string -> string
   147   val quote: string -> string
   148   val space_implode: string -> string list -> string
   149   val commas: string list -> string
   150   val commas_quote: string list -> string
   151   val cat_lines: string list -> string
   152   val space_explode: string -> string -> string list
   153   val split_lines: string -> string list
   154   val prefix_lines: string -> string -> string
   155   val untabify: string list -> string list
   156   val prefix: string -> string -> string
   157   val suffix: string -> string -> string
   158   val unprefix: string -> string -> string
   159   val unsuffix: string -> string -> string
   160   val replicate_string: int -> string -> string
   161   val translate_string: (string -> string) -> string -> string
   162 
   163   (*lists as sets -- see also Pure/General/ord_list.ML*)
   164   val member: ('b * 'a -> bool) -> 'a list -> 'b -> bool
   165   val insert: ('a * 'a -> bool) -> 'a -> 'a list -> 'a list
   166   val remove: ('b * 'a -> bool) -> 'b -> 'a list -> 'a list
   167   val subtract: ('b * 'a -> bool) -> 'b list -> 'a list -> 'a list
   168   val merge: ('a * 'a -> bool) -> 'a list * 'a list -> 'a list
   169   val mem: ''a * ''a list -> bool
   170   val mem_int: int * int list -> bool
   171   val mem_string: string * string list -> bool
   172   val union: ''a list * ''a list -> ''a list
   173   val union_int: int list * int list -> int list
   174   val union_string: string list * string list -> string list
   175   val gen_union: ('a * 'a -> bool) -> 'a list * 'a list -> 'a list
   176   val gen_inter: ('a * 'b -> bool) -> 'a list * 'b list -> 'a list
   177   val inter: ''a list * ''a list -> ''a list
   178   val inter_int: int list * int list -> int list
   179   val inter_string: string list * string list -> string list
   180   val subset: ''a list * ''a list -> bool
   181   val subset_int: int list * int list -> bool
   182   val subset_string: string list * string list -> bool
   183   val eq_set: ''a list * ''a list -> bool
   184   val eq_set_string: string list * string list -> bool
   185   val gen_subset: ('a * 'b -> bool) -> 'a list * 'b list -> bool
   186   val gen_eq_set: ('a * 'b -> bool) -> 'a list * 'b list -> bool
   187   val \ : ''a list * ''a -> ''a list
   188   val \\ : ''a list * ''a list -> ''a list
   189   val distinct: ('a * 'a -> bool) -> 'a list -> 'a list
   190   val duplicates: ('a * 'a -> bool) -> 'a list -> 'a list
   191   val has_duplicates: ('a * 'a -> bool) -> 'a list -> bool
   192 
   193   (*lists as multisets*)
   194   val remove1: ('b * 'a -> bool) -> 'b -> 'a list -> 'a list
   195   val submultiset: ('a * 'b -> bool) -> 'a list * 'b list -> bool
   196 
   197   (*balanced trees*)
   198   exception Balance
   199   val fold_bal: ('a * 'a -> 'a) -> 'a list -> 'a
   200   val access_bal: ('a -> 'a) * ('a -> 'a) * 'a -> int -> int -> 'a
   201   val accesses_bal: ('a -> 'a) * ('a -> 'a) * 'a -> int -> 'a list
   202 
   203   (*orders*)
   204   val is_equal: order -> bool
   205   val rev_order: order -> order
   206   val make_ord: ('a * 'a -> bool) -> 'a * 'a -> order
   207   val int_ord: int * int -> order
   208   val string_ord: string * string -> order
   209   val fast_string_ord: string * string -> order
   210   val option_ord: ('a * 'b -> order) -> 'a option * 'b option -> order
   211   val prod_ord: ('a * 'b -> order) -> ('c * 'd -> order) -> ('a * 'c) * ('b * 'd) -> order
   212   val dict_ord: ('a * 'b -> order) -> 'a list * 'b list -> order
   213   val list_ord: ('a * 'b -> order) -> 'a list * 'b list -> order
   214   val sort: ('a * 'a -> order) -> 'a list -> 'a list
   215   val sort_distinct: ('a * 'a -> order) -> 'a list -> 'a list
   216   val sort_strings: string list -> string list
   217   val sort_wrt: ('a -> string) -> 'a list -> 'a list
   218 
   219   (*random numbers*)
   220   exception RANDOM
   221   val random: unit -> real
   222   val random_range: int -> int -> int
   223   val one_of: 'a list -> 'a
   224   val frequency: (int * 'a) list -> 'a
   225 
   226   (*current directory*)
   227   val cd: string -> unit
   228   val pwd: unit -> string
   229 
   230   (*misc*)
   231   val divide_and_conquer: ('a -> 'a list * ('b list -> 'b)) -> 'a -> 'b
   232   val partition_eq: ('a * 'a -> bool) -> 'a list -> 'a list list
   233   val partition_list: (int -> 'a -> bool) -> int -> int -> 'a list -> 'a list list
   234   val gensym: string -> string
   235   type stamp
   236   val stamp: unit -> stamp
   237   type serial
   238   val serial: unit -> serial
   239   val serial_string: unit -> string
   240   structure Object: sig type T end
   241 end;
   242 
   243 signature LIBRARY =
   244 sig
   245   include BASIC_LIBRARY
   246   val foldl: ('a * 'b -> 'a) -> 'a * 'b list -> 'a
   247   val foldr: ('a * 'b -> 'b) -> 'a list * 'b -> 'b
   248   val take: int * 'a list -> 'a list
   249   val drop: int * 'a list -> 'a list
   250   val last_elem: 'a list -> 'a
   251 end;
   252 
   253 structure Library: LIBRARY =
   254 struct
   255 
   256 (* functions *)
   257 
   258 fun I x = x;
   259 fun K x = fn _ => x;
   260 fun flip f x y = f y x;
   261 fun curry f x y = f (x, y);
   262 fun uncurry f (x, y) = f x y;
   263 
   264 (*application and structured results -- old version*)
   265 fun (x, y) |>>> f = let val (x', z) = f x in (x', (y, z)) end;
   266 
   267 (*conditional application*)
   268 fun b ? f = fn x => if b then f x else x;
   269 
   270 (*composition with multiple args*)
   271 fun (f oo g) x y = f (g x y);
   272 fun (f ooo g) x y z = f (g x y z);
   273 fun (f oooo g) x y z w = f (g x y z w);
   274 
   275 (*function exponentiation: f(...(f x)...) with n applications of f*)
   276 fun funpow n f x =
   277   let fun rep (0, x) = x
   278         | rep (n, x) = rep (n - 1, f x)
   279   in rep (n, x) end;
   280 
   281 
   282 (* exceptions *)
   283 
   284 val do_transform_failure = ref true;
   285 
   286 fun transform_failure exn f x =
   287   if ! do_transform_failure then
   288     f x handle Interrupt => raise Interrupt | e => raise exn e
   289   else f x;
   290 
   291 exception EXCEPTION of exn * string;
   292 
   293 datatype 'a result =
   294   Result of 'a |
   295   Exn of exn;
   296 
   297 fun capture f x = Result (f x) handle e => Exn e;
   298 
   299 fun release (Result y) = y
   300   | release (Exn e) = raise e;
   301 
   302 fun get_result (Result x) = SOME x
   303   | get_result _ = NONE;
   304 
   305 fun get_exn (Exn exn) = SOME exn
   306   | get_exn _ = NONE;
   307 
   308 
   309 (* errors *)
   310 
   311 exception SYS_ERROR of string;
   312 fun sys_error msg = raise SYS_ERROR msg;
   313 
   314 exception ERROR of string;
   315 fun error msg = raise ERROR msg;
   316 
   317 fun cat_error "" msg = error msg
   318   | cat_error msg1 msg2 = error (msg1 ^ "\n" ^ msg2);
   319 
   320 fun assert_all pred list msg =
   321   let
   322     fun ass [] = ()
   323       | ass (x :: xs) = if pred x then ass xs else error (msg x);
   324   in ass list end;
   325 
   326 
   327 (* pairs *)
   328 
   329 fun pair x y = (x, y);
   330 fun rpair x y = (y, x);
   331 
   332 fun fst (x, y) = x;
   333 fun snd (x, y) = y;
   334 
   335 fun eq_fst eq ((x1, _), (x2, _)) = eq (x1, x2);
   336 fun eq_snd eq ((_, y1), (_, y2)) = eq (y1, y2);
   337 fun eq_pair eqx eqy ((x1, y1), (x2, y2)) = eqx (x1, x2) andalso eqy (y1, y2);
   338 
   339 fun swap (x, y) = (y, x);
   340 
   341 fun apfst f (x, y) = (f x, y);
   342 fun apsnd f (x, y) = (x, f y);
   343 fun pairself f (x, y) = (f x, f y);
   344 
   345 
   346 (* booleans *)
   347 
   348 (*polymorphic equality*)
   349 fun equal x y = x = y;
   350 fun not_equal x y = x <> y;
   351 
   352 (*combining predicates*)
   353 fun p orf q = fn x => p x orelse q x;
   354 fun p andf q = fn x => p x andalso q x;
   355 
   356 (*exists pred [x1, ..., xn] ===> pred x1 orelse ... orelse pred xn*)
   357 fun exists (pred: 'a -> bool) : 'a list -> bool =
   358   let fun boolf [] = false
   359         | boolf (x :: xs) = pred x orelse boolf xs
   360   in boolf end;
   361 
   362 (*forall pred [x1, ..., xn] ===> pred x1 andalso ... andalso pred xn*)
   363 fun forall (pred: 'a -> bool) : 'a list -> bool =
   364   let fun boolf [] = true
   365         | boolf (x :: xs) = pred x andalso boolf xs
   366   in boolf end;
   367 
   368 
   369 (* flags *)
   370 
   371 fun set flag = (flag := true; true);
   372 fun reset flag = (flag := false; false);
   373 fun toggle flag = (flag := not (! flag); ! flag);
   374 
   375 fun change r f = r := f (! r);
   376 
   377 (*temporarily set flag during execution*)
   378 fun setmp flag value f x =
   379   let
   380     val orig_value = ! flag;
   381     val _ = flag := value;
   382     val result = capture f x;
   383     val _ = flag := orig_value;
   384   in release result end;
   385 
   386 
   387 
   388 (** lists **)
   389 
   390 exception UnequalLengths;
   391 
   392 fun single x = [x];
   393 
   394 fun the_single [x] = x
   395   | the_single _ = raise Empty;
   396 
   397 fun singleton f x = the_single (f [x]);
   398 
   399 fun apply [] x = x
   400   | apply (f :: fs) x = apply fs (f x);
   401 
   402 
   403 (* fold -- old versions *)
   404 
   405 (*the following versions of fold are designed to fit nicely with infixes*)
   406 
   407 (*  (op @) (e, [x1, ..., xn])  ===>  ((e @ x1) @ x2) ... @ xn
   408     for operators that associate to the left (TAIL RECURSIVE)*)
   409 fun foldl (f: 'a * 'b -> 'a) : 'a * 'b list -> 'a =
   410   let fun itl (e, [])  = e
   411         | itl (e, a::l) = itl (f(e, a), l)
   412   in  itl end;
   413 
   414 (*  (op @) ([x1, ..., xn], e)  ===>   x1 @ (x2 ... @ (xn @ e))
   415     for operators that associate to the right (not tail recursive)*)
   416 fun foldr f (l, e) =
   417   let fun itr [] = e
   418         | itr (a::l) = f(a, itr l)
   419   in  itr l  end;
   420 
   421 (*  (op @) [x1, ..., xn]  ===>   x1 @ (x2 ... @ (x[n-1] @ xn))
   422     for n > 0, operators that associate to the right (not tail recursive)*)
   423 fun foldr1 f [] = raise Empty
   424   | foldr1 f l =
   425       let fun itr [x] = x
   426             | itr (x::l) = f(x, itr l)
   427       in  itr l  end;
   428 
   429 fun foldl_map f =
   430   let
   431     fun fold_aux (x, []) = (x, [])
   432       | fold_aux (x, y :: ys) =
   433           let
   434             val (x', y') = f (x, y);
   435             val (x'', ys') = fold_aux (x', ys);
   436           in (x'', y' :: ys') end;
   437   in fold_aux end;
   438 
   439 
   440 (* basic list functions *)
   441 
   442 fun eq_list eq (list1, list2) =
   443   let
   444     fun eq_lst (x :: xs, y :: ys) = eq (x, y) andalso eq_lst (xs, ys)
   445       | eq_lst _ = true;
   446   in length list1 = length list2 andalso eq_lst (list1, list2) end;
   447 
   448 fun maps f [] = []
   449   | maps f (x :: xs) = f x @ maps f xs;
   450 
   451 fun chop 0 xs = ([], xs)
   452   | chop _ [] = ([], [])
   453   | chop n (x :: xs) = chop (n - 1) xs |>> cons x;
   454 
   455 (*take the first n elements from a list*)
   456 fun take (n, []) = []
   457   | take (n, x :: xs) =
   458       if n > 0 then x :: take (n - 1, xs) else [];
   459 
   460 (*drop the first n elements from a list*)
   461 fun drop (n, []) = []
   462   | drop (n, x :: xs) =
   463       if n > 0 then drop (n - 1, xs) else x :: xs;
   464 
   465 fun dropwhile P [] = []
   466   | dropwhile P (ys as x::xs) = if P x then dropwhile P xs else ys;
   467 
   468 (*return nth element of a list, where 0 designates the first element;
   469   raise Subscript if list too short*)
   470 fun nth xs i = List.nth (xs, i);
   471 
   472 fun nth_list xss i = nth xss i handle Subscript => [];
   473 
   474 fun nth_map 0 f (x :: xs) = f x :: xs
   475   | nth_map n f (x :: xs) = x :: nth_map (n - 1) f xs
   476   | nth_map _ _ [] = raise Subscript;
   477 
   478 fun map_index f =
   479   let
   480     fun mapp _ [] = []
   481       | mapp i (x :: xs) = f (i, x) :: mapp (i+1) xs
   482   in mapp 0 end;
   483 
   484 fun fold_index f =
   485   let
   486     fun fold_aux _ [] y = y
   487       | fold_aux i (x :: xs) y = fold_aux (i+1) xs (f (i, x) y);
   488   in fold_aux 0 end;
   489 
   490 val last_elem = List.last;
   491 
   492 (*rear decomposition*)
   493 fun split_last [] = raise Empty
   494   | split_last [x] = ([], x)
   495   | split_last (x :: xs) = apfst (cons x) (split_last xs);
   496 
   497 (*find the position of an element in a list*)
   498 fun find_index pred =
   499   let fun find _ [] = ~1
   500         | find n (x :: xs) = if pred x then n else find (n + 1) xs;
   501   in find 0 end;
   502 
   503 fun find_index_eq x = find_index (equal x);
   504 
   505 (*find first element satisfying predicate*)
   506 val find_first = List.find;
   507 
   508 (*get first element by lookup function*)
   509 fun get_first _ [] = NONE
   510   | get_first f (x :: xs) =
   511       (case f x of
   512         NONE => get_first f xs
   513       | some => some);
   514 
   515 fun get_index f =
   516   let
   517     fun get _ [] = NONE
   518       | get i (x :: xs) =
   519           case f x
   520            of NONE => get (i + 1) xs
   521             | SOME y => SOME (i, y)
   522   in get 0 end;
   523 
   524 val flat = List.concat;
   525 
   526 fun unflat (xs :: xss) ys =
   527       let val (ps, qs) = chop (length xs) ys
   528       in ps :: unflat xss qs end
   529   | unflat [] [] = []
   530   | unflat _ _ = raise UnequalLengths;
   531 
   532 fun burrow f xss = unflat xss (f (flat xss));
   533 
   534 fun fold_burrow f xss s =
   535   apfst (unflat xss) (f (flat xss) s);
   536 
   537 (*separate s [x1, x2, ..., xn]  ===>  [x1, s, x2, s, ..., s, xn]*)
   538 fun separate s (x :: (xs as _ :: _)) = x :: s :: separate s xs
   539   | separate _ xs = xs;
   540 
   541 (*make the list [x, x, ..., x] of length n*)
   542 fun replicate n (x: 'a) : 'a list =
   543   let fun rep (0, xs) = xs
   544         | rep (n, xs) = rep (n - 1, x :: xs)
   545   in
   546     if n < 0 then raise Subscript
   547     else rep (n, [])
   548   end;
   549 
   550 fun translate_string f = String.translate (f o String.str);
   551 
   552 (*multiply [a, b, c, ...] * [xs, ys, zs, ...]*)
   553 fun multiply [] _ = []
   554   | multiply (x :: xs) yss = map (cons x) yss @ multiply xs yss;
   555 
   556 (*direct product*)
   557 fun product _ [] = []
   558   | product [] _ = []
   559   | product (x :: xs) ys = map (pair x) ys @ product xs ys;
   560 
   561 
   562 (* filter *)
   563 
   564 (*copy the list preserving elements that satisfy the predicate*)
   565 val filter = List.filter;
   566 fun filter_out f = filter (not o f);
   567 val map_filter = List.mapPartial;
   568 
   569 
   570 (* lists of pairs *)
   571 
   572 exception UnequalLengths;
   573 
   574 fun map2 _ [] [] = []
   575   | map2 f (x :: xs) (y :: ys) = f x y :: map2 f xs ys
   576   | map2 _ _ _ = raise UnequalLengths;
   577 
   578 fun fold2 f [] [] z = z
   579   | fold2 f (x :: xs) (y :: ys) z = fold2 f xs ys (f x y z)
   580   | fold2 f _ _ _ = raise UnequalLengths;
   581 
   582 fun zip_options (x :: xs) (SOME y :: ys) = (x, y) :: zip_options xs ys
   583   | zip_options (_ :: xs) (NONE :: ys) = zip_options xs ys
   584   | zip_options _ [] = []
   585   | zip_options [] _ = raise UnequalLengths;
   586 
   587 (*combine two lists forming a list of pairs:
   588   [x1, ..., xn] ~~ [y1, ..., yn]  ===>  [(x1, y1), ..., (xn, yn)]*)
   589 fun [] ~~ [] = []
   590   | (x :: xs) ~~ (y :: ys) = (x, y) :: (xs ~~ ys)
   591   | _ ~~ _ = raise UnequalLengths;
   592 
   593 (*inverse of ~~; the old 'split':
   594   [(x1, y1), ..., (xn, yn)]  ===>  ([x1, ..., xn], [y1, ..., yn])*)
   595 val split_list = ListPair.unzip;
   596 
   597 
   598 (* prefixes, suffixes *)
   599 
   600 fun is_prefix _ [] _ = true
   601   | is_prefix eq (x :: xs) (y :: ys) = eq (x, y) andalso is_prefix eq xs ys
   602   | is_prefix eq _ _ = false;
   603 
   604 (* [x1, ..., xi, ..., xn]  --->  ([x1, ..., x(i-1)], [xi, ..., xn])
   605    where xi is the first element that does not satisfy the predicate*)
   606 fun take_prefix (pred : 'a -> bool)  (xs: 'a list) : 'a list * 'a list =
   607   let fun take (rxs, []) = (rev rxs, [])
   608         | take (rxs, x :: xs) =
   609             if  pred x  then  take(x :: rxs, xs)  else  (rev rxs, x :: xs)
   610   in  take([], xs)  end;
   611 
   612 fun chop_prefix eq ([], ys) = ([], ([], ys))
   613   | chop_prefix eq (xs, []) = ([], (xs, []))
   614   | chop_prefix eq (xs as x :: xs', ys as y :: ys') =
   615       if eq (x, y) then
   616         let val (ps', xys'') = chop_prefix eq (xs', ys')
   617         in (x :: ps', xys'') end
   618       else ([], (xs, ys));
   619 
   620 (* [x1, ..., xi, ..., xn]  --->  ([x1, ..., xi], [x(i+1), ..., xn])
   621    where xi is the last element that does not satisfy the predicate*)
   622 fun take_suffix _ [] = ([], [])
   623   | take_suffix pred (x :: xs) =
   624       (case take_suffix pred xs of
   625         ([], sffx) => if pred x then ([], x :: sffx) else ([x], sffx)
   626       | (prfx, sffx) => (x :: prfx, sffx));
   627 
   628 fun prefixes1 [] = []
   629   | prefixes1 (x :: xs) = map (cons x) ([] :: prefixes1 xs);
   630 
   631 fun prefixes xs = [] :: prefixes1 xs;
   632 
   633 fun suffixes1 xs = map rev (prefixes1 (rev xs));
   634 fun suffixes xs = [] :: suffixes1 xs;
   635 
   636 
   637 
   638 (** integers **)
   639 
   640 fun inc i = (i := ! i + 1; ! i);
   641 fun dec i = (i := ! i - 1; ! i);
   642 
   643 
   644 (* lists of integers *)
   645 
   646 (*make the list [from, from + 1, ..., to]*)
   647 fun (i upto j) =
   648   if i > j then [] else i :: (i + 1 upto j);
   649 
   650 (*make the list [from, from - 1, ..., to]*)
   651 fun (i downto j) =
   652   if i < j then [] else i :: (i - 1 downto j);
   653 
   654 
   655 (* convert integers to strings *)
   656 
   657 (*expand the number in the given base;
   658   example: radixpand (2, 8) gives [1, 0, 0, 0]*)
   659 fun radixpand (base, num) : int list =
   660   let
   661     fun radix (n, tail) =
   662       if n < base then n :: tail
   663       else radix (n div base, (n mod base) :: tail)
   664   in radix (num, []) end;
   665 
   666 (*expands a number into a string of characters starting from "zerochar";
   667   example: radixstring (2, "0", 8) gives "1000"*)
   668 fun radixstring (base, zerochar, num) =
   669   let val offset = ord zerochar;
   670       fun chrof n = chr (offset + n)
   671   in implode (map chrof (radixpand (base, num))) end;
   672 
   673 
   674 val string_of_int = Int.toString;
   675 
   676 fun signed_string_of_int i =
   677   if i < 0 then "-" ^ string_of_int (~ i) else string_of_int i;
   678 
   679 fun string_of_indexname (a, 0) = a
   680   | string_of_indexname (a, i) = a ^ "_" ^ string_of_int i;
   681 
   682 
   683 (* read integers *)
   684 
   685 fun read_intinf radix cs =
   686   let
   687     val zero = ord "0";
   688     val limit = zero + radix;
   689     fun scan (num, []) = (num, [])
   690       | scan (num, c :: cs) =
   691         if zero <= ord c andalso ord c < limit then
   692           scan (IntInf.fromInt radix * num + IntInf.fromInt (ord c - zero), cs)
   693         else (num, c :: cs);
   694   in scan (IntInf.fromInt 0, cs) end;
   695 
   696 fun read_int cs = apfst IntInf.toInt (read_intinf 10 cs);
   697 
   698 fun oct_char s = chr (IntInf.toInt (#1 (read_intinf 8 (explode s))));
   699 
   700 
   701 
   702 (** strings **)
   703 
   704 (* functions tuned for strings, avoiding explode *)
   705 
   706 fun nth_string str i =
   707   (case try String.substring (str, i, 1) of
   708     SOME s => s
   709   | NONE => raise Subscript);
   710 
   711 fun fold_string f str x0 =
   712   let
   713     val n = size str;
   714     fun iter (x, i) =
   715       if i < n then iter (f (String.substring (str, i, 1)) x, i + 1) else x;
   716   in iter (x0, 0) end;
   717 
   718 fun exists_string pred str =
   719   let
   720     val n = size str;
   721     fun ex i = i < n andalso (pred (String.substring (str, i, 1)) orelse ex (i + 1));
   722   in ex 0 end;
   723 
   724 fun forall_string pred = not o exists_string (not o pred);
   725 
   726 (*enclose in brackets*)
   727 fun enclose lpar rpar str = lpar ^ str ^ rpar;
   728 fun unenclose str = String.substring (str, 1, size str - 2);
   729 
   730 (*simple quoting (does not escape special chars)*)
   731 val quote = enclose "\"" "\"";
   732 
   733 (*space_implode "..." (explode "hello") = "h...e...l...l...o"*)
   734 fun space_implode a bs = implode (separate a bs);
   735 
   736 val commas = space_implode ", ";
   737 val commas_quote = commas o map quote;
   738 
   739 (*concatenate messages, one per line, into a string*)
   740 val cat_lines = space_implode "\n";
   741 
   742 (*space_explode "." "h.e..l.lo" = ["h", "e", "", "l", "lo"]*)
   743 fun space_explode _ "" = []
   744   | space_explode sep s = String.fields (fn c => str c = sep) s;
   745 
   746 val split_lines = space_explode "\n";
   747 
   748 fun prefix_lines "" txt = txt
   749   | prefix_lines prfx txt = txt |> split_lines |> map (fn s => prfx ^ s) |> cat_lines;
   750 
   751 fun untabify chs =
   752   let
   753     val tab_width = 8;
   754 
   755     fun untab pos [] ys = rev ys
   756       | untab pos ("\n" :: xs) ys = untab 0 xs ("\n" :: ys)
   757       | untab pos ("\t" :: xs) ys =
   758           let val d = tab_width - (pos mod tab_width)
   759           in untab (pos + d) xs (replicate d " " @ ys) end
   760       | untab pos (c :: xs) ys = untab (pos + 1) xs (c :: ys);
   761   in
   762     if not (exists (fn c => c = "\t") chs) then chs
   763     else untab 0 chs []
   764   end;
   765 
   766 fun prefix prfx s = prfx ^ s;
   767 fun suffix sffx s = s ^ sffx;
   768 
   769 fun unprefix prfx s =
   770   if String.isPrefix prfx s then String.substring (s, size prfx, size s - size prfx)
   771   else raise Fail "unprefix";
   772 
   773 fun unsuffix sffx s =
   774   if String.isSuffix sffx s then String.substring (s, 0, size s - size sffx)
   775   else raise Fail "unsuffix";
   776 
   777 fun replicate_string 0 _ = ""
   778   | replicate_string 1 a = a
   779   | replicate_string k a =
   780       if k mod 2 = 0 then replicate_string (k div 2) (a ^ a)
   781       else replicate_string (k div 2) (a ^ a) ^ a;
   782 
   783 
   784 
   785 (** lists as sets -- see also Pure/General/ord_list.ML **)
   786 
   787 (*canonical member, insert, remove*)
   788 fun member eq list x =
   789   let
   790     fun memb [] = false
   791       | memb (y :: ys) = eq (x, y) orelse memb ys;
   792   in memb list end;
   793 
   794 fun insert eq x xs = if member eq xs x then xs else x :: xs;
   795 fun remove eq x xs = if member eq xs x then filter_out (fn y => eq (x, y)) xs else xs;
   796 
   797 fun subtract eq = fold (remove eq);
   798 
   799 fun merge _ ([], ys) = ys
   800   | merge eq (xs, ys) = fold_rev (insert eq) ys xs;
   801 
   802 (*old-style infixes*)
   803 fun x mem xs = member (op =) xs x;
   804 fun (x: int) mem_int xs = member (op =) xs x;
   805 fun (x: string) mem_string xs = member (op =) xs x;
   806 
   807 
   808 (*union of sets represented as lists: no repetitions*)
   809 fun xs union [] = xs
   810   | [] union ys = ys
   811   | (x :: xs) union ys = xs union (insert (op =) x ys);
   812 
   813 (*union of sets, optimized version for ints*)
   814 fun (xs:int list) union_int [] = xs
   815   | [] union_int ys = ys
   816   | (x :: xs) union_int ys = xs union_int (insert (op =) x ys);
   817 
   818 (*union of sets, optimized version for strings*)
   819 fun (xs:string list) union_string [] = xs
   820   | [] union_string ys = ys
   821   | (x :: xs) union_string ys = xs union_string (insert (op =) x ys);
   822 
   823 (*generalized union*)
   824 fun gen_union eq (xs, []) = xs
   825   | gen_union eq ([], ys) = ys
   826   | gen_union eq (x :: xs, ys) = gen_union eq (xs, insert eq x ys);
   827 
   828 
   829 (*intersection*)
   830 fun [] inter ys = []
   831   | (x :: xs) inter ys =
   832       if x mem ys then x :: (xs inter ys) else xs inter ys;
   833 
   834 (*intersection, optimized version for ints*)
   835 fun ([]:int list) inter_int ys = []
   836   | (x :: xs) inter_int ys =
   837       if x mem_int ys then x :: (xs inter_int ys) else xs inter_int ys;
   838 
   839 (*intersection, optimized version for strings *)
   840 fun ([]:string list) inter_string ys = []
   841   | (x :: xs) inter_string ys =
   842       if x mem_string ys then x :: (xs inter_string ys) else xs inter_string ys;
   843 
   844 (*generalized intersection*)
   845 fun gen_inter eq ([], ys) = []
   846   | gen_inter eq (x::xs, ys) =
   847       if member eq ys x then x :: gen_inter eq (xs, ys)
   848       else gen_inter eq (xs, ys);
   849 
   850 
   851 (*subset*)
   852 fun [] subset ys = true
   853   | (x :: xs) subset ys = x mem ys andalso xs subset ys;
   854 
   855 (*subset, optimized version for ints*)
   856 fun ([]: int list) subset_int ys = true
   857   | (x :: xs) subset_int ys = x mem_int ys andalso xs subset_int ys;
   858 
   859 (*subset, optimized version for strings*)
   860 fun ([]: string list) subset_string ys = true
   861   | (x :: xs) subset_string ys = x mem_string ys andalso xs subset_string ys;
   862 
   863 (*set equality*)
   864 fun eq_set (xs, ys) =
   865   xs = ys orelse (xs subset ys andalso ys subset xs);
   866 
   867 (*set equality for strings*)
   868 fun eq_set_string ((xs: string list), ys) =
   869   xs = ys orelse (xs subset_string ys andalso ys subset_string xs);
   870 
   871 fun gen_subset eq (xs, ys) = forall (member eq ys) xs;
   872 
   873 fun gen_eq_set eq (xs, ys) =
   874   eq_list eq (xs, ys) orelse
   875     (gen_subset eq (xs, ys) andalso gen_subset (eq o swap) (ys, xs));
   876 
   877 
   878 (*removing an element from a list WITHOUT duplicates*)
   879 fun (y :: ys) \ x = if x = y then ys else y :: (ys \ x)
   880   | [] \ x = [];
   881 fun ys \\ xs = foldl (op \) (ys,xs);
   882 
   883 
   884 (*makes a list of the distinct members of the input; preserves order, takes
   885   first of equal elements*)
   886 fun distinct eq lst =
   887   let
   888     fun dist (rev_seen, []) = rev rev_seen
   889       | dist (rev_seen, x :: xs) =
   890           if member eq rev_seen x then dist (rev_seen, xs)
   891           else dist (x :: rev_seen, xs);
   892   in dist ([], lst) end;
   893 
   894 (*returns a list containing all repeated elements exactly once; preserves
   895   order, takes first of equal elements*)
   896 fun duplicates eq lst =
   897   let
   898     fun dups (rev_dups, []) = rev rev_dups
   899       | dups (rev_dups, x :: xs) =
   900           if member eq rev_dups x orelse not (member eq xs x) then
   901             dups (rev_dups, xs)
   902           else dups (x :: rev_dups, xs);
   903   in dups ([], lst) end;
   904 
   905 fun has_duplicates eq =
   906   let
   907     fun dups [] = false
   908       | dups (x :: xs) = member eq xs x orelse dups xs;
   909   in dups end;
   910 
   911 
   912 
   913 (** lists as multisets **)
   914 
   915 fun remove1 _ _ [] = raise Empty
   916   | remove1 eq y (x::xs) = if eq (y, x) then xs else x :: remove1 eq y xs;
   917 
   918 fun submultiset _ ([], _)  = true
   919   | submultiset eq (x :: xs, ys) = member eq ys x andalso submultiset eq (xs, remove1 eq x ys);
   920 
   921 
   922 
   923 (** balanced trees **)
   924 
   925 exception Balance;      (*indicates non-positive argument to balancing fun*)
   926 
   927 (*balanced folding; avoids deep nesting*)
   928 fun fold_bal f [x] = x
   929   | fold_bal f [] = raise Balance
   930   | fold_bal f xs =
   931       let val (ps, qs) = chop (length xs div 2) xs
   932       in  f (fold_bal f ps, fold_bal f qs)  end;
   933 
   934 (*construct something of the form f(...g(...(x)...)) for balanced access*)
   935 fun access_bal (f, g, x) n i =
   936   let fun acc n i =     (*1<=i<=n*)
   937           if n=1 then x else
   938           let val n2 = n div 2
   939           in  if i<=n2 then f (acc n2 i)
   940                        else g (acc (n-n2) (i-n2))
   941           end
   942   in  if 1<=i andalso i<=n then acc n i else raise Balance  end;
   943 
   944 (*construct ALL such accesses; could try harder to share recursive calls!*)
   945 fun accesses_bal (f, g, x) n =
   946   let fun acc n =
   947           if n=1 then [x] else
   948           let val n2 = n div 2
   949               val acc2 = acc n2
   950           in  if n-n2=n2 then map f acc2 @ map g acc2
   951                          else map f acc2 @ map g (acc (n-n2)) end
   952   in  if 1<=n then acc n else raise Balance  end;
   953 
   954 
   955 
   956 (** orders **)
   957 
   958 fun is_equal EQUAL = true
   959   | is_equal _ = false;
   960 
   961 fun rev_order LESS = GREATER
   962   | rev_order EQUAL = EQUAL
   963   | rev_order GREATER = LESS;
   964 
   965 (*assume rel is a linear strict order*)
   966 fun make_ord rel (x, y) =
   967   if rel (x, y) then LESS
   968   else if rel (y, x) then GREATER
   969   else EQUAL;
   970 
   971 val int_ord = Int.compare;
   972 val string_ord = String.compare;
   973 
   974 fun fast_string_ord (s1, s2) =
   975   (case int_ord (size s1, size s2) of EQUAL => string_ord (s1, s2) | ord => ord);
   976 
   977 fun option_ord ord (SOME x, SOME y) = ord (x, y)
   978   | option_ord _ (NONE, NONE) = EQUAL
   979   | option_ord _ (NONE, SOME _) = LESS
   980   | option_ord _ (SOME _, NONE) = GREATER;
   981 
   982 (*lexicographic product*)
   983 fun prod_ord a_ord b_ord ((x, y), (x', y')) =
   984   (case a_ord (x, x') of EQUAL => b_ord (y, y') | ord => ord);
   985 
   986 (*dictionary order -- in general NOT well-founded!*)
   987 fun dict_ord elem_ord (x :: xs, y :: ys) =
   988       (case elem_ord (x, y) of EQUAL => dict_ord elem_ord (xs, ys) | ord => ord)
   989   | dict_ord _ ([], []) = EQUAL
   990   | dict_ord _ ([], _ :: _) = LESS
   991   | dict_ord _ (_ :: _, []) = GREATER;
   992 
   993 (*lexicographic product of lists*)
   994 fun list_ord elem_ord (xs, ys) =
   995   (case int_ord (length xs, length ys) of EQUAL => dict_ord elem_ord (xs, ys) | ord => ord);
   996 
   997 
   998 (* sorting *)
   999 
  1000 (*quicksort -- stable, i.e. does not reorder equal elements*)
  1001 fun quicksort unique ord =
  1002   let
  1003     fun qsort [] = []
  1004       | qsort (xs as [_]) = xs
  1005       | qsort (xs as [x, y]) =
  1006           (case ord (x, y) of
  1007             LESS => xs
  1008           | EQUAL => if unique then [x] else xs
  1009           | GREATER => [y, x])
  1010       | qsort xs =
  1011           let val (lts, eqs, gts) = part (nth xs (length xs div 2)) xs
  1012           in qsort lts @ eqs @ qsort gts end
  1013     and part _ [] = ([], [], [])
  1014       | part pivot (x :: xs) = add (ord (x, pivot)) x (part pivot xs)
  1015     and add LESS x (lts, eqs, gts) = (x :: lts, eqs, gts)
  1016       | add EQUAL x (lts, [], gts) = (lts, [x], gts)
  1017       | add EQUAL x (res as (lts, eqs, gts)) = if unique then res else (lts, x :: eqs, gts)
  1018       | add GREATER x (lts, eqs, gts) = (lts, eqs, x :: gts);
  1019   in qsort end;
  1020 
  1021 fun sort ord = quicksort false ord;
  1022 fun sort_distinct ord = quicksort true ord;
  1023 
  1024 val sort_strings = sort string_ord;
  1025 fun sort_wrt sel xs = sort (string_ord o pairself sel) xs;
  1026 
  1027 
  1028 
  1029 (** random numbers **)
  1030 
  1031 exception RANDOM;
  1032 
  1033 fun rmod x y = x - y * Real.realFloor (x / y);
  1034 
  1035 local
  1036   val a = 16807.0;
  1037   val m = 2147483647.0;
  1038   val random_seed = ref 1.0;
  1039 in
  1040 
  1041 fun random () =
  1042   let val r = rmod (a * !random_seed) m
  1043   in (random_seed := r; r) end;
  1044 
  1045 end;
  1046 
  1047 fun random_range l h =
  1048   if h < l orelse l < 0 then raise RANDOM
  1049   else l + Real.floor (rmod (random ()) (real (h - l + 1)));
  1050 
  1051 fun one_of xs = nth xs (random_range 0 (length xs - 1));
  1052 
  1053 fun frequency xs =
  1054   let
  1055     val sum = foldl op + (0, map fst xs);
  1056     fun pick n ((k: int, x) :: xs) =
  1057       if n <= k then x else pick (n - k) xs
  1058   in pick (random_range 1 sum) xs end;
  1059 
  1060 
  1061 
  1062 (** current directory **)
  1063 
  1064 val cd = OS.FileSys.chDir;
  1065 val pwd = OS.FileSys.getDir;
  1066 
  1067 
  1068 
  1069 (** misc **)
  1070 
  1071 fun divide_and_conquer decomp x =
  1072   let val (ys, recomb) = decomp x
  1073   in recomb (map (divide_and_conquer decomp) ys) end;
  1074 
  1075 
  1076 (*Partition a list into buckets  [ bi, b(i+1), ..., bj ]
  1077    putting x in bk if p(k)(x) holds.  Preserve order of elements if possible.*)
  1078 fun partition_list p i j =
  1079   let fun part k xs =
  1080             if k>j then
  1081               (case xs of [] => []
  1082                          | _ => raise Fail "partition_list")
  1083             else
  1084             let val (ns, rest) = List.partition (p k) xs;
  1085             in  ns :: part(k+1)rest  end
  1086   in  part i end;
  1087 
  1088 fun partition_eq (eq:'a * 'a -> bool) =
  1089   let
  1090     fun part [] = []
  1091       | part (x :: ys) =
  1092           let val (xs, xs') = List.partition (fn y => eq (x, y)) ys
  1093           in (x::xs)::(part xs') end
  1094   in part end;
  1095 
  1096 
  1097 
  1098 (* generating identifiers *)
  1099 
  1100 (** Freshly generated identifiers; supplied prefix MUST start with a letter **)
  1101 local
  1102 (*Maps 0-61 to A-Z, a-z, 0-9; exclude _ or ' to avoid clash with internal/unusual indentifiers*)
  1103 fun gensym_char i =
  1104   if i<26 then chr (ord "A" + i)
  1105   else if i<52 then chr (ord "a" + i - 26)
  1106   else chr (ord "0" + i - 52);
  1107 
  1108 val char_vec = Vector.tabulate (62, gensym_char);
  1109 fun newid n = implode (map (fn i => Vector.sub (char_vec, i)) (radixpand (62, n)));
  1110 
  1111 val gensym_seed = ref 0;
  1112 
  1113 in
  1114   fun gensym pre = pre ^ newid (inc gensym_seed);
  1115 end;
  1116 
  1117 
  1118 (* stamps and serial numbers *)
  1119 
  1120 type stamp = unit ref;
  1121 val stamp: unit -> stamp = ref;
  1122 
  1123 type serial = int;
  1124 local val count = ref 0
  1125 in fun serial () = inc count end;
  1126 
  1127 val serial_string = string_of_int o serial;
  1128 
  1129 
  1130 (* generic objects *)
  1131 
  1132 (*note that the builtin exception datatype may be extended by new
  1133   constructors at any time*)
  1134 structure Object = struct type T = exn end;
  1135 
  1136 end;
  1137 
  1138 structure BasicLibrary: BASIC_LIBRARY = Library;
  1139 open BasicLibrary;